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orthomodular lattice

Orthogonality Relations

Let LLL be an orthocomplemented lattice and a,b∈LabLa,b\in L. aaa is said to be orthogonal to bbb if a≤b⟂asuperscriptbperpendicular-toa\leq b^{{\perp}}, denoted by a⟂bperpendicular-toaba\perp b. If a≤b⟂asuperscriptbperpendicular-toa\leq b^{{\perp}}, then b=b⟂⁣⟂≤a⟂bsuperscriptbperpendicular-toperpendicular-tosuperscriptaperpendicular-tob=b^{{\perp\perp}}\leq a^{{\perp}}, so ⟂perpendicular-to\perp is a symmetric relation on LLL. It is easy to see that, for any a,b∈LabLa,b\in L, a⟂bperpendicular-toaba\perp bimpliesa∧b=0ab0a\wedge b=0, and a⟂a⟂perpendicular-toasuperscriptaperpendicular-toa\perp a^{{\perp}}.

For any a∈LaLa\in L, define M⁢(a):={c∈L∣c⟂a⁢ and ⁢1=c∨a}assignMaconditional-setcLperpendicular-toca and 1caM(a):=\{c\in L\mid c\perp a\mbox{ and }1=c\vee a\}. An element of M⁢(a)MaM(a) is called an orthogonal complement of aaa. We have a⟂∈M⁢(a)superscriptaperpendicular-toMaa^{{\perp}}\in M(a), and any orthogonal complement of aaa is a complement of aaa.

If we replace the 111 in M⁢(a)MaM(a) by an arbitrary element b≥abab\geq a, then we have the set

An element of M⁢(a,b)MabM(a,b) is called an orthogonal complement of aaarelative tobbb. Clearly, M⁢(a)=M⁢(a,1)MaMa1M(a)=M(a,1). Also, for a,c≤bacba,c\leq b, c∈M⁢(a,b)cMabc\in M(a,b)iffa∈M⁢(c,b)aMcba\in M(c,b). As a result, we can define a symmetricbinaryoperator⊕direct-sum\oplus on [0,b]0b[0,b], given by b=a⊕cbdirect-sumacb=a\oplus c iff c∈M⁢(a,b)cMabc\in M(a,b). Note that b=b⊕0bdirect-sumb0b=b\oplus 0.

Before the main definition, we define one more operation: b-a:=b∧a⟂assignbabsuperscriptaperpendicular-tob-a:=b\wedge a^{{\perp}}. Some properties: (1) a-a=0aa0a-a=0, a-0=aa0aa-0=a, 0-a=00a00-a=0, a-1=0a10a-1=0, and 1-a=a⟂1asuperscriptaperpendicular-to1-a=a^{{\perp}}; (2) b-a=a⟂-b⟂basuperscriptaperpendicular-tosuperscriptbperpendicular-tob-a=a^{{\perp}}-b^{{\perp}}; and (3) if a≤baba\leq b, then a⟂(b-a)perpendicular-toabaa\perp(b-a) and a⊕(b-a)≤bdirect-sumababa\oplus(b-a)\leq b.

Definition

The orthomodular law can be restated as follows: if x≤yxyx\leq y, then
y=x∨(y∧x⟂)yxysuperscriptxperpendicular-toy=x\vee(y\wedge x^{{\perp}}). Equivalently, x≤yxyx\leq y implies y=(y∧x)∨(y∧x⟂)yyxysuperscriptxperpendicular-toy=(y\wedge x)\vee(y\wedge x^{{\perp}}). Note that the equation is automatically true in an arbitrary distributive lattice, even without the assumption that x≤yxyx\leq y.

An nice example of an orthomodular lattice that is not modular can be found in the reference below.

Remarks.

Orthomodular lattices were first studied by John von Neumann and Garett Birkhoff, when they were trying to develop the logic of quantum mechanics by studying the structure of the lattice ℙ⁢(H)ℙH\mathbb{P}(H) of projection operators on a Hilbert space HHH. However, the term was coined by Irving Kaplansky, when it was realized that ℙ⁢(H)ℙH\mathbb{P}(H), while orthocomplemented, is not modular. Rather, it satisfies a variant of the modular law as indicated above.